Begin With What Is Given
A direct proof usually has a clear starting point: the hypotheses of the statement. In Working Backward From a Conclusion, we examined a goal to see what would be enough to establish it. Here we focus on the other direction of planning. Start with the information known to be true, ask what follows from it, and continue until one of those consequences is the conclusion or can be connected to it.
For a conditional statement, the proof does not begin by assuming its conclusion. It begins by taking arbitrary objects from the stated domain and supposing the hypothesis holds. A proof then justifies each subsequent line using definitions, algebra, or results established earlier in the course. The resulting chain has the form $$ \text{hypotheses}\Longrightarrow \text{consequences}\Longrightarrow \text{conclusion}. $$
The word “forward” describes the direction of justification, not a requirement to make an obvious next move. A hypothesis may have several useful consequences. The task is to choose one that preserves the assumptions and helps reach the goal. In the previous tutorial, the conclusion suggested what evidence would be useful. Forward reasoning supplies that evidence from facts already available.
Extract Consequences Carefully
A hypothesis often has a useful form once it is translated. If an integer is even, its definition gives a factorization \(n=2k\) for some integer \(k\). If a real number satisfies \(x>c\), subtracting \(c\) gives \(x-c>0\). If two quantities are ordered, adding the same real number to both preserves their order, by the order property established earlier in this course.
Each translation gives a possible next line, not necessarily the final step. For instance, knowing \(x-c>0\) can help establish that a factor is positive. Knowing \(n=2k\) can help exhibit an even integer or a divisibility relation. A good proof keeps track of why the new fact is useful and whether all the conditions required to use it are satisfied.
For a universal claim, let the variable be arbitrary in the required set, such as \(\mathbb R\) or \(\mathbb Z\).
Record the exact inequalities, equalities, or definitions supplied by the statement.
Use definitions, valid algebra, or earlier results; check signs and domain restrictions along the way.
Identify the implication or definition that turns what is known into the conclusion.
Make the final connection explicit so the reader can see that the original claim has been proved.
Build a Chain From an Inequality
Suppose the hypothesis is an inequality. Adding or subtracting the same real number on both sides can produce a more useful comparison. Sometimes the consequence tells us the sign of a factor, and that sign makes multiplication valid in the desired direction. The sign check is part of the reasoning: multiplying an inequality by a positive number preserves its direction, whereas multiplying by a negative number reverses it.
Worked Example: Deriving a Polynomial Inequality From a Lower Bound
Theorem. If \(t>1\), then \(t^2+t>2\).
Forward plan. The hypothesis gives \(t-1>0\). It also gives \(t+2>3>0\). These two facts imply that their product is positive. Expanding that product yields the desired expression minus \(2\).
Proof. Let \(t\in\mathbb R\) and suppose \(t>1\). Subtracting \(1\) gives \(t-1>0\). Adding \(2\) gives \(t+2>3>0\). The product of two positive real numbers is positive, so $$ (t-1)(t+2)>0. $$ Expanding the left side gives \(t^2+t-2>0\). Adding \(2\) to both sides yields \(t^2+t>2\), as required.
The proof moves from the hypothesis to two sign facts, then to a product inequality, and finally to the stated conclusion. Notice that the expansion is an equality, while the positivity comes from the two separately established signs. Keeping those roles distinct makes the reasoning easy to check.
A proof need not spell out every elementary arithmetic detail at the same length, but it should not omit a step whose validity depends on a condition. In particular, if an argument multiplies inequalities, divides by an expression, or uses a definition involving a special domain, state or establish the relevant condition first.
Definitions Turn Hypotheses Into Evidence
Definitions are a common source of forward steps. They turn an abstract description into a form that can be used in a calculation. When a hypothesis says that an integer has a property defined by a representation, introduce the integer in that representation and carry it through the argument. When the conclusion itself is a definition, the final step must verify every part of that definition, including that any proposed object belongs to the required domain.
Worked Example: Showing That Consecutive Integers Have an Even Product
Theorem. For every integer \(n\), the product \(n(n+1)\) is even.
Forward plan. An integer is either even or odd. In the even case, write \(n=2k\). In the odd case, write \(n=2k+1\). Each representation makes one of the consecutive factors visibly even. The case distinction covers every integer.
Proof. Let \(n\in\mathbb Z\). If \(n\) is even, then by the definition of evenness there is an integer \(k\) such that \(n=2k\). Thus $$ n(n+1)=2k(n+1). $$ Since \(k\) and \(n+1\) are integers, \(k(n+1)\) is an integer. The displayed equation expresses \(n(n+1)\) as \(2\) times an integer, so \(n(n+1)\) is even.
If \(n\) is not even, then \(n\) is odd. By the definition of oddness, there is an integer \(k\) such that \(n=2k+1\). Therefore $$ n(n+1)=(2k+1)(2k+2)=2(2k+1)(k+1). $$ Both \(2k+1\) and \(k+1\) are integers, so their product is an integer. Hence this expression is \(2\) times an integer, and \(n(n+1)\) is even in this case as well. The two cases exhaust the integers, so the theorem holds for every \(n\in\mathbb Z\).
The forward argument starts with an arbitrary integer, rather than assuming the product is even. In each case the hypothesis supplies a representation, and the calculation produces exactly the form required by the definition of evenness.
Conditions Can Be Part of the Argument
A hypothesis can do more than provide an expression to manipulate. It can make an operation legitimate. For example, a statement that \(x>0\) tells us that \(x\ne0\), so division by \(x\) is allowed. It also tells us that dividing an inequality by \(x\) preserves its direction. Without the positivity hypothesis, division would require separate attention to whether the divisor is positive, negative, or zero.
Worked Example: Using Positivity Before Dividing
Theorem. If \(x>0\), then \(x+\dfrac{1}{x}\geq2\).
Forward plan. The square \((x-1)^2\) is nonnegative. The hypothesis \(x>0\) allows us to divide this inequality by \(x\) without changing its direction. Expanding then produces the desired expression.
Proof. Let \(x\in\mathbb R\) and suppose \(x>0\). The square of a real number is nonnegative, so $$ (x-1)^2\geq0. $$ Because \(x>0\), we have \(x\ne0\), and division by \(x\) preserves the inequality: $$ \frac{(x-1)^2}{x}\geq0. $$ Expanding the numerator and dividing each term by the positive \(x\) gives $$ x-2+\frac{1}{x}\geq0. $$ Adding \(2\) to both sides yields \(x+\dfrac{1}{x}\geq2\), which proves the claim.
The assumption \(x>0\) has two jobs here: it ensures the reciprocal is defined and it determines how division affects the inequality. If \(x\) were allowed to be negative, dividing by \(x\) would reverse the inequality; if \(x=0\), the reciprocal would not be defined. The stated hypothesis handles both issues.
| What the hypothesis gives | Forward consequence | Condition to keep in view |
|---|---|---|
| An integer is even | Write it as \(2k\) for some \(k\in\mathbb Z\). | Verify that the new factor is still an integer. |
| A real number satisfies \(x>c\) | Subtract \(c\) to obtain \(x-c>0\). | Do not infer the sign of an unrelated expression without a reason. |
| A real number satisfies \(x>0\) | Conclude \(x\ne0\) and use positivity when multiplying or dividing. | State the sign-dependent rule being used. |
| Two quantities satisfy \(a<b\) | Add the same quantity to both sides to preserve the order. | Multiplying by a quantity requires knowing its sign. |
Keep the Proof Moving Toward Its Goal
A forward proof can become a list of true facts without becoming an argument. After deriving a consequence, ask what it contributes. Does it provide a required factorization? Establish a sign? Produce the integer or real number that a definition asks for? If a fact has no apparent connection to the conclusion, it may not be the useful next step.
The chain does not have to be long. In the consecutive-integers example, the case representation immediately supplies a factor of \(2\). In the inequality example, positivity of two factors supplies a positive product, and expansion identifies that product with the expression needed. The goal is not to generate as many consequences as possible; it is to find a justified route from the assumptions to the endpoint.
It is also useful to separate what is assumed from what has been proved. A proof may begin, “Suppose \(x>0\),” but it may not simply write down \(x+\frac1x\geq2\) as a starting fact when that is the claim under consideration. The square inequality supplies a new fact; the hypothesis supplies the sign condition; together they justify the next step. This keeps the proof from quietly relying on its own conclusion.
In Combining Forward and Backward Reasoning, these two planning directions will be used together: the hypotheses indicate what can be derived, and the conclusion indicates what kind of fact would complete the proof. For now, keep their roles distinct. A forward proof begins with what is given and makes every connection explicit.
Check Your Understanding
For each question, focus on which facts are available at that point in the proof and why the next step is valid.
- If a proof begins with an arbitrary integer \(n\) that is even, what representation does the definition provide, and what domain must the new variable belong to?
- In the proof that \(t>1\) implies \(t^2+t>2\), which two positive factors produce the needed product inequality?
- Why does the proof that \(n(n+1)\) is even need two cases, and how does each case produce a factor of \(2\)?
- In the reciprocal inequality example, what two distinct roles does the assumption \(x>0\) play?
- If a proof derives a true statement from its hypothesis but never connects that statement to the conclusion, what is missing?
- Why is writing the desired conclusion as the first line of a proof not a valid way to reason forward from the hypotheses?